Let me begin by explaining the power dynamics of Fascist Italy first.

As you may very well already know, Italy was technically still a Monarchy and Mussolini a Prime Minister; but his powers and those of the King had become more and more balanced over time and the two were, in a sense, 'forced' to co-operate as neither could have gotten rid of the other by himself; Mussolini wanted to overthrow Victor Emmanuel III sooner or later, but that would have probably lead to civil war. Both the Army and the Navy were loyal to the King; indeed, they had sworn loyalty to him and not to Mussolini. ^1 However, he had the militia (M.V.S.N.) and the Party - whose gerarchi (most prominent members) sat in the Grand Council of Fascism - behind him.

Now, it seems obvious that any successful attempt at getting rid of one party by the other would have needed the intervention of either an external force or someone who could've tipped the balance of power; Victor Emmanuel's attempted coup of March 1940, ^2 in which he sought to replace a more and more war-like Mussolini with a loyal Galeazzo Ciano, couldn't be carried through because (in part) of the Grand Council's opposition to such plans.

However, by 1943 the situation had changed - Mussolini was now rather impopular even among his own as his actions had clearly brought Italy to ruin; the invasion of Sicily and the Allied bombing of Rome a few days earlier proved to be the last straw. Both the Grand Council and the King now shared the same goal, and an exhautorated Mussolini was in no position to resist: on July 25th, 1943, the Council approved the so-called ordine del giorno Grandi which was effectively a vote of no-confidence... Mussolini learnt of this and asked the King to receive him at the Quirinal.

And after a brief meeting, he was sacked and arrested.

[...] Finally, weren't the council afraid of Mussolini? Didn't Il Duce have the power to get rid of the members that opposed him?

Not at the time... but they came to fear him again after he'd been freed by the Germans and put in charge of the Italian Social Republic; many of the gerarchi who had passed the vote of no-confidence would have later been executed. The M.V.S.N. was disbanded, the Duce replaced by the more ineffective Pietro Badoglio and Army soldiers were left to fend for themselves (Badoglio didn't give clear orders, and the chain of command was now disrupted).

It should also be noted that Mussolini was never - due to his role as PM - as powerful as Hitler or Stalin ever were; yes, he got rid of many dissidents and anti-Fascists but all of this still happened in a somewhat-constitutional setting in that they had to be trialled by a dedicated tribunal and either sent into exile (al confino) on remote islands - such as S. Stefano - or, sometimes, sentenced to death. But turning against the King or his own party (like Stalin did) was not possible because 1) such a course of action was objectively beyond Mussolini's reach and 2) at that point, both parties would have probably worked together to oust him as it happened in July 1943.

 

^1 VV. AA. - Manuale per il Graduato, p. 9 (Tipografia del Senato, Roma 1934 - Anno XII).

The oath every new recruit in the Regio Esercito had to take recited:

"I swear to be loyal to the King and his Royal successors, to respect and faithfully observe the Statute [that'd be the Statuto Albertino, Italy's pre-war constitution] as well as the other laws of the State, and to fulfil my duties towards the country with no other goal than the good of the King and that of the Fatherland, which are one and the same".

^2 F. Perfetti, L. Cavicchioli et al. - Nuova Storia Contemporanea, n. 5 (ed. Le Lettere, Sept/Oct. 2002).